Visualizing the real and complex roots of [math]ax^2+bx+c=0[/math]. [br][br]When the graph of [math]y=ax^2+bx+c[/math] intersects the x-axis, the roots are real and we can visualize them on the graph as x-intercepts.[br][br]But what about when there are no real roots, i.e. when the graph does not intersect the x-axis? The equation still has 2 roots, but now they are complex. This visual imagines the cartesian graph floating above the real (or x-axis) of the complex plane. In this manner, real roots correspond with traditional x-intercepts, but now we can see some of the symmetry in how the complex roots relate to the original graph.[br][br](Note: Despite showing complex roots, this plot makes sense only for the real-valued function [math]f(x) = ax^2+bx+c[/math]. In other words, only for input values, x, that result in real values of [math]f(x)[/math].)

[b]Question[/b]: Why are the complex roots always the points where grey curve hits the complex plane?